"Are we talking just a few mW, or several hundred mW?"
Somewhere in there... +19dBm, about 79mW, is a pretty typical maximum output power for cards I'm finding on Google for 802.11g.
The actual power output changes with time depending on the needed signal strengths.
I found some routers listed as doing 400mW.
If you want actual hard limits, I think you're going to have to dig in the IEEE 802.11 specifications.
"What kind of internal antenna does such a notebook/netbook PC have in order to communicate with a Wi-Fi router/network? "
I took apart a T41 thinkpad with a dead screen, and it had a pair of small printed circuit antennas.
As luck would have it, they're selling some on eBay:http://tinyurl.com/dylmub
The other one is a little different.
The other one is mounted vertically along the side of the screen, and the one in the picture is mounted horizontally along the top, and presumably the WiFi transceiver uses a diversity system to select the better one.
There's a huge variety of antennas in wireless devices; the routers probably use pretty conventional collinear types, but I haven't really taken many of those apart :-) A lot of PCMCIA laptop cards (before they built antennas into every laptop) used printed circuit antennas right on the main board of the card.